1 00:00:00,000 --> 00:00:04,200 Hello everyone. Welcome to Week 9 
part 2 of Carmen's Core Concepts, 2 00:00:04,200 --> 00:00:05,933 my name is Carmen Bruni, 3 00:00:05,933 --> 00:00:10,166 and in this video series, we talk about 
the weekly concepts in Math 135. 4 00:00:10,166 --> 00:00:10,800 5 00:00:10,800 --> 00:00:13,933 So part 1 of Week 9, that was RSA. 6 00:00:13,933 --> 00:00:16,400 In part 2, we're going to talk 
about complex numbers. 7 00:00:16,400 --> 00:00:20,133 Okay so what is a complex 
number? So a complex number 8 00:00:20,133 --> 00:00:23,866 is an expression of the form x plus 
y i, where x and y are real and 9 00:00:23,866 --> 00:00:26,266 i is the imaginary unit. 10 00:00:26,266 --> 00:00:29,500 So i on its own has no meaning. i is just 11 00:00:29,500 --> 00:00:30,433 12 00:00:30,433 --> 00:00:32,166 an object, okay? 13 00:00:32,166 --> 00:00:32,733 14 00:00:32,733 --> 00:00:36,666 Denote the set of complex 
numbers by the set 15 00:00:36,666 --> 00:00:40,066 as follows: x plus y i where 
x and y are real numbers. 16 00:00:40,066 --> 00:00:44,700 Some examples are 1 plus 2i, 
3i, square root of 13 plus pi i, 17 00:00:44,700 --> 00:00:47,800 2, which is the same as 2 plus 0i. 18 00:00:47,800 --> 00:00:51,000 So something to think about is that the real numbers are 19 00:00:51,000 --> 00:00:52,833 contained inside the complex numbers, 20 00:00:52,833 --> 00:00:55,733 so are the rational numbers, so are 
the integers and so on and so forth. 21 00:00:55,733 --> 00:00:57,000 22 00:00:57,000 --> 00:00:59,266 So a lot of people 23 00:00:59,266 --> 00:01:04,366 when they first see complex numbers, they 
define i as the square root of minus 1. 24 00:01:04,366 --> 00:01:08,000 I'm not going to do this and there's a lot of 
good reasons why we shouldn't be doing this, 25 00:01:08,000 --> 00:01:10,633 and one of the reasons is that, 26 00:01:10,633 --> 00:01:11,533 27 00:01:11,533 --> 00:01:13,533 that doesn't really make sense, 
right? I mean when we 28 00:01:13,533 --> 00:01:16,100 say like the square root of 
4 what we mean is the 29 00:01:16,100 --> 00:01:18,466 positive number such that 30 00:01:18,466 --> 00:01:21,233 when I multiply it by itself I get 2… 31 00:01:21,233 --> 00:01:24,266 or I get 4, sorry, which is 2. 32 00:01:24,266 --> 00:01:25,900 33 00:01:25,900 --> 00:01:28,633 But I don't want to define i like that 34 00:01:28,633 --> 00:01:32,166 because it's not a positive number, it’s… 35 00:01:32,166 --> 00:01:34,366 it's an object, okay? 36 00:01:34,366 --> 00:01:37,466 So if all I'm doing is saying 
that i is some object, 37 00:01:37,466 --> 00:01:39,833 then what am I going to do 
with this? Well we'll see. 38 00:01:39,833 --> 00:01:43,333 At the moment, right now, C is 
just a set of things, it's formed… 39 00:01:43,333 --> 00:01:44,266 40 00:01:44,266 --> 00:01:48,000 this plus sign is actually - has 
no meaning as of now, okay, 41 00:01:48,000 --> 00:01:49,600 it's just really a symbol. 42 00:01:49,600 --> 00:01:52,000 It's denoted by x with the plus symbol 43 00:01:52,000 --> 00:01:54,366 and then y times i, okay? 44 00:01:54,366 --> 00:01:57,466 Now what we're going to do is we're going 
to give the plus sign some meaning, 45 00:01:57,466 --> 00:02:00,000 and we're going to give 
multiplication some meaning. 46 00:02:00,000 --> 00:02:02,666 So we're going to turn the complex numbers into a ring. 47 00:02:02,666 --> 00:02:03,533 48 00:02:03,533 --> 00:02:06,833 The last note: 2 complex 
numbers, z equals x plus y i and 49 00:02:06,833 --> 00:02:11,600 w equals u plus v i are equal if and 
only if x equals u and y equals v. 50 00:02:11,600 --> 00:02:13,900 51 00:02:13,900 --> 00:02:16,933 Let's turn C into a ring and a field, okay? 52 00:02:16,933 --> 00:02:17,966 53 00:02:17,966 --> 00:02:20,900 So we turn C into a commutative ring by defining 54 00:02:20,900 --> 00:02:21,866 55 00:02:21,866 --> 00:02:25,066 an addition and actually 
a subtraction operation, 56 00:02:25,066 --> 00:02:28,866 and how do we do this? So 
we're going to define x plus y i 57 00:02:28,866 --> 00:02:31,666 plus or minus u plus v i to be 58 00:02:31,666 --> 00:02:34,066 x plus or minus u, as a real number, 59 00:02:34,066 --> 00:02:37,600 plus y plus or minus v, as 
a real number, times i. 60 00:02:37,600 --> 00:02:40,366 And that way we get addition and we get subtraction. 61 00:02:40,366 --> 00:02:44,166 Notice that this also defines 
the additive inverse, right? 62 00:02:44,166 --> 00:02:47,833 So we have the addition structure, 
and the multiplication structure: 63 00:02:47,833 --> 00:02:51,766 x plus y i times u plus v i, that's going to be defined to be 64 00:02:51,766 --> 00:02:54,466 x u minus y v 65 00:02:54,466 --> 00:02:56,966 plus x v plus 66 00:02:56,966 --> 00:02:59,933 u y all multiplied by i. 67 00:02:59,933 --> 00:03:03,533 That is the structure that 
we are giving our set C, 68 00:03:03,533 --> 00:03:06,500 and this will make it a commutative ring. Okay? 69 00:03:06,500 --> 00:03:08,666 70 00:03:08,666 --> 00:03:11,400 Something to note now, this multiplication operation above, 71 00:03:11,400 --> 00:03:16,166 well if we had 0 plus i times 
0 plus i, we can see that 72 00:03:16,166 --> 00:03:19,366 the left hand side would 
be denoted by i squared, 73 00:03:19,366 --> 00:03:23,733 and the right hand side would tell us that 74 00:03:23,733 --> 00:03:25,866 the value is minus 1. 75 00:03:25,866 --> 00:03:26,500 76 00:03:26,500 --> 00:03:30,700 So this object that we defined with these 
two addition and multiplication operations, 77 00:03:30,700 --> 00:03:35,300 tell us that this new object, i, has the 
property that i squared is minus 1. 78 00:03:35,300 --> 00:03:37,433 So something to note 
is now that we have… 79 00:03:37,433 --> 00:03:39,633 we now have solutions to 
the polynomial, let's say, 80 00:03:39,633 --> 00:03:44,000 x squared plus 1 equals 0 over the 
complex numbers. The solution is i, 81 00:03:44,000 --> 00:03:46,800 actually it’s plus or minus i, both of those are valid. 82 00:03:46,800 --> 00:03:48,466 83 00:03:48,466 --> 00:03:49,900 So something to think about, okay? 84 00:03:49,900 --> 00:03:52,033 So this is one of the reasons - 
oh I didn't mention the motivation 85 00:03:52,033 --> 00:03:53,733 for defining complex numbers, 86 00:03:53,733 --> 00:03:58,466 but one of the motivations is to 
try to give algebraic expressions… 87 00:03:58,466 --> 00:03:59,700 roots. 88 00:03:59,700 --> 00:04:02,433 So to try to give all algebraic 
expression solutions, 89 00:04:02,433 --> 00:04:06,366 and we'll see this in the upcoming 
weeks why this actually does this. 90 00:04:06,366 --> 00:04:07,633 91 00:04:07,633 --> 00:04:10,566 Now that we know that i squared is 
minus 1, we can actually remember this 92 00:04:10,566 --> 00:04:13,533 operation here, the 
multiplication operation, just 93 00:04:13,533 --> 00:04:15,433 by multiplying it out 
how we would before. 94 00:04:15,433 --> 00:04:19,000 We would multiply x times u, and then 
we would take x times v i and then 95 00:04:19,000 --> 00:04:22,333 y i times u and then y i times v i. 96 00:04:22,333 --> 00:04:25,233 And if we do that, we 
actually see that the… 97 00:04:25,233 --> 00:04:27,166 under the… 98 00:04:27,166 --> 00:04:30,566 under the new fact that i squared 
is minus 1, we now see that 99 00:04:30,566 --> 00:04:34,133 this y v i squared goes back to the 100 00:04:34,133 --> 00:04:36,366 x u minus y v, 101 00:04:36,366 --> 00:04:40,133 and then x v plus u y i is 
the second term there. 102 00:04:40,133 --> 00:04:41,200 103 00:04:41,200 --> 00:04:44,433 So here we see that multiplication is actually something that we know here, 104 00:04:44,433 --> 00:04:48,166 and addition is not too hard either. It’s basically common factoring the i. 105 00:04:48,166 --> 00:04:49,433 106 00:04:49,433 --> 00:04:52,400 So addition and multiplication are 
actually very natural operations here. 107 00:04:52,400 --> 00:04:53,400 108 00:04:53,400 --> 00:04:56,933 We note that C is a field, so how 
do we note that it's a field? So 109 00:04:56,933 --> 00:04:59,700 we know that the inverse of 
any non-zero complex number, 110 00:04:59,700 --> 00:05:02,733 x plus y i, is actually defined as x over 111 00:05:02,733 --> 00:05:07,666 x squared plus y squared, minus y 
over x squared plus y squared, times i. 112 00:05:07,666 --> 00:05:10,666 If you multiply this by x plus y i, 113 00:05:10,666 --> 00:05:12,833 you will see that the answer is 1. 114 00:05:12,833 --> 00:05:15,266 I will leave that as an exercise 
though for you to check out. 115 00:05:15,266 --> 00:05:17,100 116 00:05:17,100 --> 00:05:20,833 This is the hard way to think about 
the inverse. We'll learn in a bit 117 00:05:20,833 --> 00:05:23,433 that we can look at inverses using 118 00:05:23,433 --> 00:05:26,166 an operation called the 
modulus of a complex number, 119 00:05:26,166 --> 00:05:29,500 and that will help us to remember 
the inverse formula here. 120 00:05:29,500 --> 00:05:30,200 121 00:05:30,200 --> 00:05:32,333 But for now, let's just - 
I'm just going to state it 122 00:05:32,333 --> 00:05:34,000 and I'll leave it as an exercise for you to check, 123 00:05:34,000 --> 00:05:36,700 that if I multiply by x plus y i I get 1. 124 00:05:36,700 --> 00:05:38,833 125 00:05:38,833 --> 00:05:41,900 Complex conjugation, this is a 
very useful operation but… 126 00:05:41,900 --> 00:05:42,300 127 00:05:42,300 --> 00:05:44,766 before - I mean if you just look 
at it, it seems very simple. 128 00:05:44,766 --> 00:05:48,600 A complex conjugation of a 
complex number z equals x plus y i 129 00:05:48,600 --> 00:05:52,333 is z bar, which is defined to 
be x minus y i. So we just 130 00:05:52,333 --> 00:05:55,466 negate the imaginary component. 131 00:05:55,466 --> 00:05:59,066 So if you think about this 
graphically, if you have a vector… 132 00:05:59,066 --> 00:06:02,133 or a complex number in the 133 00:06:02,133 --> 00:06:04,000 real-imaginary plane, 134 00:06:04,000 --> 00:06:06,566 the complex conjugate is 
just its reflection about 135 00:06:06,566 --> 00:06:10,033 the line y equals x, so something 
like that is what it would look like. 136 00:06:10,033 --> 00:06:11,733 137 00:06:11,733 --> 00:06:14,200 Properties of conjugates, 
these are very easy to prove 138 00:06:14,200 --> 00:06:18,233 so I'm only going to prove a 
couple here. They're not hard, 139 00:06:18,233 --> 00:06:21,266 one way to do this is to just 
introduce coordinates, so write z as 140 00:06:21,266 --> 00:06:24,900 x plus y i, write w as u plus v i 141 00:06:24,900 --> 00:06:27,466 and then just go through the 
operations. So add them, 142 00:06:27,466 --> 00:06:31,166 then take the conjugate, is the same as taking the conjugate and then adding them, 143 00:06:31,166 --> 00:06:33,266 and so on and so forth. 144 00:06:33,266 --> 00:06:35,500 145 00:06:35,500 --> 00:06:39,233 What's this one supposed to 
be? Is it z z bar? Or z bar… 146 00:06:39,233 --> 00:06:42,900 ah there should be a multiplication 
sign here, but we can't see it. 147 00:06:42,900 --> 00:06:45,733 The conjugate of z times 
w is supposed to be the 148 00:06:45,733 --> 00:06:48,000 the conjugate of z times the conjugate of w. 149 00:06:48,000 --> 00:06:52,766 I'll be sure to correct this on the thing. 
I think there's a…no, I don't know. 150 00:06:52,766 --> 00:06:55,600 I mean there should be a multiplication 
sign, that's really what we want. 151 00:06:55,600 --> 00:06:57,066 152 00:06:57,066 --> 00:07:01,766 If I do the process of complex 
conjugation twice, I get z. 153 00:07:01,766 --> 00:07:04,633 z plus z bar, so z plus its 
conjugate, is 2 times the real part, 154 00:07:04,633 --> 00:07:09,933 and z minus its conjugate is 2 times the imagin- 2i times the imaginary part. 155 00:07:09,933 --> 00:07:12,300 So again, all these can be done 
with just introducing coordinates. 156 00:07:12,300 --> 00:07:15,400 So here's an example of 3, 
for example: z bar bar is 157 00:07:15,400 --> 00:07:20,000 x plus y i bar bar, which 
is the bar of x minus y i, 158 00:07:20,000 --> 00:07:22,500 which again - so if we're taking 
the minus sign and negating it, 159 00:07:22,500 --> 00:07:24,566 then we just get plus, and 
that's the same as z. 160 00:07:24,566 --> 00:07:25,333 161 00:07:25,333 --> 00:07:28,133 Similarly with addition and subtraction. 162 00:07:28,133 --> 00:07:29,300 163 00:07:29,300 --> 00:07:31,000 So if you introduce 
coordinates, it's not too hard. 164 00:07:31,000 --> 00:07:33,666 By coordinates here, 
I mean the x plus y i. 165 00:07:33,666 --> 00:07:34,433 166 00:07:34,433 --> 00:07:36,833 And this is what we call the standard form of a complex number. 167 00:07:36,833 --> 00:07:39,833 We'll see at the end the 
polar form is coming up. 168 00:07:39,833 --> 00:07:41,933 So that’s properties of conjugates here.